
The upper bound of \((\lambda _n)_{n \in \mathbb {N}}\)  in Theorem REF  can be written as \(\dfrac{1-\epsilon ^{\prime }}{\mu +\Vert K\Vert }\) , where \(\epsilon ^{\prime } \in ]0,1[.\)  Indeed, for \(\epsilon <\frac{1}{(1-\epsilon ^{\prime })^2}-1\) , we get \(\lambda _n<\frac{1}{\sqrt{1+\epsilon }(\mu +\Vert K\Vert )}.\)

In the deterministic setting,
the convergence rate of the primal-dual gap for structure convex optimization involving infimal convolutions were also investigated in [1]}, [2]}. Furthermore, in the deterministic case and \(\alpha _n=0\)  \(\forall n \in \mathbb {N}\) , our result is one in [2]}. The convergence rate \(\mathcal {O}(1/ N)\)  of the primal-dual gap also obtain in [4]}.

In the stochastic setting, our result is the same convergence rate of the primal-dual gap as in [5]}, [6]}.


